High Order Strong Stability Preserving Time Discretizations
High Order Strong Stability Preserving Time Discretizations
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DOI：
10.1007/s10915-008-9239-z
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发表时间：
2009-03
影响因子：
2.5
通讯作者：
S. Gottlieb;D. Ketcheson;Chi-Wang Shu
中科院分区：
文献类型：
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作者：
S. Gottlieb;D. Ketcheson;Chi-Wang Shu
为了保证具有间断解的双曲型偏微分方程数值解的非线性稳定性，提出了一种强保稳定的高阶时间离散方法. SSP方法保持强稳定性的属性，在任何范数，凸函数的空间离散耦合一阶欧拉时间步进。本文介绍了SSP方法的发展和强稳定性保持的时间步长限制和收缩性之间的联系。数值例子表明，常见的线性稳定，但不强的稳定性保持时间离散可能会导致违反重要的有界性属性，而SSP方法保证所需的属性，只要这些属性满足向前欧拉时间步长。我们回顾了线性和非线性问题的最优显式和隐式SSP Runge-Kutta和多步法。我们还讨论了光谱延迟校正方法的SSP性质。
Strong stability preserving (SSP) high order time discretizations were developed to ensure nonlinear stability properties necessary in the numerical solution of hyperbolic partial differential equations with discontinuous solutions. SSP methods preserve the strong stability properties—in any norm, seminorm or convex functional—of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the connections between the timestep restrictions for strong stability preservation and contractivity. Numerical examples demonstrate that common linearly stable but not strong stability preserving time discretizations may lead to violation of important boundedness properties, whereas SSP methods guarantee the desired properties provided only that these properties are satisfied with forward Euler timestepping. We review optimal explicit and implicit SSP Runge–Kutta and multistep methods, for linear and nonlinear problems. We also discuss the SSP properties of spectral deferred correction methods.